Although some ideas of calculus had been recognised for centuries, modern calculus was developed simultaneously and independently by Liebniz (Germany) and Newton (England). Some people believe that Liebniz developed the mathematical aspects whereas Newton was more effective in applying the newly discovered techniques to physics.
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Calculus; by a long shot.
Just about all of calculus is based on differential and integral calculus, including Calculus 1! However, Calculus 1 is more likely to cover differential calculus, with integral calculus soon after. So there really isn't a right answer for this question.
It is certainly used in calculus, just as calculus can be used in trigonometry.
Calculus is the correct spelling.
No. Calculus if a field of mathematics.